Question

The straight line $r=\hat{i}-\hat{j}+\hat{k}+\lambda 2\hat{i}+\hat{j}-\hat{k}=4$ are.

• A. Perpendicular to each other
• B. Parallel
• C. Inclined at an angle ${60}^o$
• D. Inclines at an angle ${45}^o$

Answer 1

The correct option is A Perpendicular to each other

Lines are, $r_1=\hat{i}-\hat{j}+\hat{k}$

$r_2=\widehat{2i}+\hat{j}-\hat{k}$

So we know, $\cos \theta =\frac{\overrightarrow{r_1.}\overrightarrow{r_2}}{\left|r_1\right|\left|r_2\right|}$

$\overrightarrow{r_1.}\overrightarrow{r_2}=2-1-1=0$

$\Rightarrow \cos \theta =0$

$\Rightarrow \theta ={\cos }^{-1}0$

$\Rightarrow \theta =\frac{\pi }{2}$

$\therefore$ They are perpendicular to each other.